11.3. Systems of Equations Homework

  1. For the following system of equations, express the equations as a matrix equation of the form \(\mathbf{A\,}\bm{x} = \bm{b}\).

    \[\begin{split}\left\{\begin{aligned} 7x_1 + 9x_2 + 4x_3 = 102 \\ 17x_1 + 8x_2 + 6x_3 = 171 \\ -7x_1 + -7x_2 + 10x_3 = -64 \end{aligned}\right.\end{split}\]

  2. Use the solve function from Python’s scipy.linalg module to find the solution to the previous system of equations.

  3. Apply elimination to the 2-by-3 augmented matrix \([\mathbf{A}\:\bm{b}]\). What is the triangular system \(\mathbf{U}\,\bm{x} = \bm{c}\)? What is the solution \(\bm{x}\)?

    \[\begin{split}\mathbf{A}\,\bm{x} = \begin{bmatrix} 2 & 3 \\ 4 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 1 \\ 17 \end{bmatrix}\end{split}\]

  4. For each system of equations, use Python’s numpy.linalg.matrix_rank function to determine if a solution exists. Note the rank of \(\bf{A}\). If a solution exists, use Python’s scipy.linalg.solve function to find it.

    1. \[\begin{split}\left\{\begin{aligned} x_1 + 2x_2 + 2x_3 = 1 \\ 4x_1 + 8x_2 + 9x_3 = 3 \\ 3x_2 + 2x_3 = 1 \end{aligned}\right.\end{split}\]
    2. \[\begin{split}\left\{\begin{aligned} x_1 + 2x_2 + 3x_3 = 1 \\ 3x_2 + x_3 = 2 \\ x_1 + 14x_2 + 7x_3 = 3 \end{aligned}\right.\end{split}\]
    3. \[\begin{split}\left\{\begin{aligned} 7x_1 + 7x_2 = 35 \\ 14x_1 + 7x_2 = 49 \end{aligned}\right.\end{split}\]